代写IB3K20 Financial Optimisation 2022-2023代写数据结构语言

日期：2024-04-10 12:02

IB3K20

Financial Optimisation

Exam Paper

[April] 2022-2023

[Question 1 - 40% of total marks]

The portfolio manager of UBS investment bank is willing to create a portfolio using the Markowitz portfolio allocation model. He considers three different assets (labelled as A, B and C).  He  then generates three rival scenarios (denoted by S1, S2  and S3 ) for the rates of returns of assets using the historical data to  represent three states of the economy as  presented in the following table. The associated probabilities of three scenarios are also provided in this table.

Scenarios	Probability	Asset A	Asset B	Asset C
S1: boom	0.35	0.45	1.30	1.15
S2: recession	0.40	0.45	0.85	0.93
S3: expansion	0.25	0.45	1.15	1.04

The standard deviations of assets B and Care estimated as 1.2 and 1.1, respectively. Assume that there is a perfect negative correlation between assets B and C. Let WA, WB    and Wc  denote weights of assets A, B and C, respectively.

a) The portfolio manager considers constructing a portfolio consisting of assets A, B and C. Compute the expected return and risk of the portfolio to be constructed by three assets. (8 marks)

b) The portfolio manager would like to create a portfolio by considering specific conditions. He thinks that the portfolio must consist of at most two assets and one of which much be asset A. Moreover, at most 50% of the capital should be invested on each asset and the short sale is not allowed. Formulate (but do not solve) the portfolio optimisation model that minimises the portfolio risk to achieve the expected  portfolio  return to  be at  least  12.5%.  Clearly define decision variables and briefly explain constraints, and the objective function. (10 marks)

c) Suppose that the portfolio manager now ignores the portfolio optimisation model developed in part (b). He wishes to  invest on only assets B and C.  Formulate the pure portfolio risk minimisation problem and compute the optimal investment strategy that minimises the portfolio risk subject to the sum of asset weights to be 1 and no short-sale conditions. What is the expected portfolio return and portfolio risk for the optimal investment strategy? (10 marks)

d) The portfolio manager is now concerned with the worst-case analysis for the mean-variance portfolio allocation  problem where the  portfolio  risk  is  minimised subject to the  portfolio return must be at least 2.5%, the sum of asset weights is equal to 1 and no short-sale exists. He assumes that the portfolio is constructed by only assets B and C and their rates of returns (labelled  as B  and c )  are  uncertain.  Moreover,  he  can  estimate  the  rates  of  returns  as B  = 0.25 + 0.2η   and c   = 0.5 + 0.4θ  such that (η, θ) ∈  U and the uncertainty set is given as  U = {(η, θ)| η + 2θ ≥ 5;   −2η + 3θ  ≥ 8;    η, θ ≥ 0 }.   Formulate   the   robust    portfolio management problem in view of the uncertainty set. Derive (but do not solve) the robust counterpart of the portfolio allocation problem using the duality theory. (12 marks)

[Question 2 - 35% of total marks]

a) Vodafone, a telecommunication company, expects to pay dividends of £5.5 and £5.75 per share for the next two years, respectively. After the first two years, the expected rate of growth in dividends per year is estimated as 3% for three consecutive years, and then 2.5% per year thereafter. Vodafone has just paid dividend of £4.95 per share for the current year. What is the new stock price if the required rate of return remains at 4%? (12 marks)

b) Suppose  that  Jane  has  just  been  admitted  to the  MSBA  programme  at  Warwick  Business School. She is currently planning to apply for a student loan of £25000. She wishes to repay the  loan  in  15  equal  annual  payments  each  of  which  includes  an  interest  and  principal. Calculate the annuity if the current interest rate of 5.5% per annum remains the same during this period. (8 marks)

c) Consider three bonds (labelled as X, Y, Z) with different cash flow features. Bonds X and Y are both  pure zero-coupon bonds with  1-year and 2-year  maturities,  respectively.  In addition, bonds X and Y have £90 and £4500 face values, respectively. Bond Z is a 2-year 25% coupon bond with a £1800 face value. The current market prices of bonds X, Y, and Z are £75, £3500, and £2304, respectively. Suppose that an investor wishes to apply a trading strategy today (at year 0) by short-selling 2 of bond Z and simultaneously buying 10 of bond X and 1 of bond Y, respectively.  Find the cash flows obtained by trading at each bond in years 0, 1, and 2. Show whether there is any gain or loss at each year 0, 1 and 2. (15 marks)

[Question 3 - 25% of total marks]

Bill, as the asset manager, is responsible for managing funds of AXA insurance company in the UK. He currently considers investing a capital of C(£) in six different high investment-grade quality bonds (labelled as A1, A2, A3, A4, A5, A6) to be able to pay off AXA’s liabilities over the next two years. The features of these bonds are presented in Table 1 in terms of maturity, coupon payments, face values as well as the current market prices. They assume that all bonds are widely available and can be purchased in any quantities at given prices.

Table 1: Features of different bonds

Bonds	A1	A2	A3	A4	A5	A6
Price (£) Coupon rates (£) Maturity (year) Face value (£)	97 7 1 100	95 8 1 105	105 9 1 110	117 4 2 120	118 5 2 125	111 6 2 108

Due to uncertain interest rates, they are expecting the future cash obligations to vary over the next two years. Bill generates the following scenario tree to model uncertain interest rates and liabilities over the next two years. The planning horizon of two years is represented by discrete time periods (as t = 0, 1,2) where the investment decisions are made and t = 0 represents year 0. The scenario tree consists  of  nodes  representing  different  realisations  with  certain  probabilities.  Each  node  of  the scenario tree is labelled in terms of time period and node number as (time_period, node_number). For instance, (2,4) at the top of a node of the scenario tree shows the fourth scenario realised in year 2 with branching probability of 0.7.

The annual interest rates, liabilities, and the probabilities of occurrence at each node of the scenario tree are presented in Table 2.

Table 2: Scenario tree structure

Node ID	(1,1)	(1,2)	(1,3)	(2,1)	(2,2)	(2,3)	(2,4)	(2,5)
Interest rates (%) Liabilities (£) Probabilities	2.7 13500 0.35	2.8 14000 0.4	2.5 12500 0.25	2.9 14500 0.45	3.1 15500 0.55	4.0 20000 1.0	3.4 17000 0.7	3.6 13000 0.3

Bill aims to develop a financial plan such that the firm’s expected final wealth at the end of planning horizon must be maximised by meeting their liabilities over the next two years. The remaining cash surplus, after paying the liabilities from the return received, can be reinvested at each year in a savings account with annual interest rates given in Table 2. In case of not having sufficient fund, they also plan to apply for a 1-year loan at the final time-period with the annual interest rates (given in Table 2) so that the total expected interest on the loan to be paid offat the beginning of year 3 must be minimised.

Formulate (but do not solve) the financial planning problem as a linear programming model. Briefly describe the decision variables, constraints, and objective function. (25 marks)